A max-plus based fundamental solution for a class of infinite dimensional Riccati equations

Abstract

A new fundamental solution for a specific class of infinite dimensional Riccati equations is developed. This fundamental solution is based on the max-plus dual of the dynamic programming solution operator (or semigroup) of an associated control problem. By taking the max-plus dual of this semigroup operator, the kernel of a dual-space integral operator may be obtained. This kernel is the dual-space Riccati solution propagation operator. Specific initial conditions for the Riccati equation correspond to the associated growth rates of the control problem terminal payoffs. Propagation of the solution of the Riccati equation from these initial conditions proceeds in the dual-space, via a max-plus convolution operation utilizing the aforementioned Riccati solution propagation operator.